composition series
If simple groups are the “primes” of group theory, a composition series is the way to factor a group into them. You build a chain of subgroups, each normal in the next, refined as far as possible so that no intermediate subgroup can be inserted. The simple quotients that come out are the prime factors — the composition factors — of the group.
Formally, a composition series of a group G is a finite chain 1 = G_0 ⊴ G_1 ⊴ ... ⊴ G_n = G in which each G_i is normal in G_{i+1} and each successive quotient G_{i+1}/G_i is a simple group. The simple quotients are the composition factors. The maximality is exactly the simplicity of the quotients: a quotient is simple precisely when no proper normal subgroup of G_{i+1} lies strictly between G_i and G_{i+1}.
Every finite group has a composition series, obtained by repeatedly choosing a maximal proper normal subgroup. Infinite groups may fail to have one — Z, for instance, has no composition series, since its subgroup chains never terminate in simple quotients. The deep point is that the list of composition factors is an invariant of the group, independent of how the series was built; this is the content of the Jordan-Hölder theorem.
For Z/12Z one composition series is 0 ⊂ ⟨6⟩ ⊂ ⟨3⟩ ⊂ Z/12Z, with successive quotients of orders 2, 2, 3 — all simple cyclic groups Z/2Z, Z/2Z, Z/3Z. A different series 0 ⊂ ⟨4⟩ ⊂ ⟨2⟩ ⊂ Z/12Z gives factors of orders 3, 2, 2: the same multiset, reordered.
Two composition series of Z/12Z with the same factors.